Is the Inverse Element in Modular Arithmetic Unique?

In summary: Anyway, to prove that y= z, using the fact that x has an inverse in Zn, write "x . y = 1" as "x . y = 1 . x" and multiply on the right by z to get "x . y . z = 1 . x . z= x . 1= x" (since x has an inverse). But by the associative law, x . y . z= x . (y . z)= x . 1= x. So, using xy= 1, x . y . z= 1 . z= z. We have both x and z equal to x . y . z so x= z.
  • #1
alphamu
2
0
Question: Let n be a positive integer and consider x,y,z to be elements of Zn. Prove that if x . y = 1 and x . z = 1, then y = z. (Since working in Zn the sign '.' means "multiplication modulo n".)

Conclude that if x has an inverse element in Zn, then the inverse element is unique.

Attempt: Well if x . y = 1 then this means that xy = n + 1 since were working in Zn. This means the same can be said about y . z = 1 that yz = n + 1. This suggests that z = x by substitution. I'm not sure of this is correct or if I'm going down right path here.

Other thing I thought about was multiplying x . y = 1 by z on both sides to prove y=z but I'm a bit stuck.

I'm also a bit confused about the last bit of the question where I have to conclude if x has inverse element (not entirely sure what this is) in Zn then the inverse element is unique.

Can anyone help please?Sent from my iPhone using Physics Forums
 
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  • #2
alphamu said:
Question: Let n be a positive integer and consider x,y,z to be elements of Zn. Prove that if x . y = 1 and x . z = 1, then y = z. (Since working in Zn the sign '.' means "multiplication modulo n".)

Conclude that if x has an inverse element in Zn, then the inverse element is unique.

Attempt: Well if x . y = 1 then this means that xy = n + 1 since were working in Zn. This means the same can be said about y . z = 1 that yz = n + 1.
This is NOT true. for example, 5*5= 1 (mod 12) because 5*5= 2(12+ 1). That is, 2n+ 1, NOT n+ 1.

This suggests that z = x by substitution. I'm not sure of this is correct or if I'm going down right path here.

Other thing I thought about was multiplying x . y = 1 by z on both sides to prove y=z but I'm a bit stuck.
Why "a bit stuck"? Where did you get stuck? Since multiplication is commutative, yes, if xy= 1 then zxy= (xz)y= z so y= z.

I'm also a bit confused about the last bit of the question where I have to conclude if x has inverse element (not entirely sure what this is) in Zn then the inverse element is unique.
An "inverse element" of x is another element, y, such that xy= yx= 1. For any "other inverse element", z, xz= 1.

Can anyone help please?


Sent from my iPhone using Physics Forums
I'm a bit puzzled why, on recognizing that you did not know the definitions well ("not entirely sure what this is"), you did not immediately look up and review the definitions.
 

Related to Is the Inverse Element in Modular Arithmetic Unique?

1. What is multiplication modulo algebra?

Multiplication modulo algebra is a mathematical operation that involves performing multiplication on integers or polynomials and then taking the remainder when dividing by a given number or polynomial. It is often used in cryptography and number theory.

2. How is multiplication modulo algebra different from regular multiplication?

In regular multiplication, the result is always an integer or a polynomial. However, in multiplication modulo algebra, the result is always a remainder, which is typically a smaller number than the original operands. It also follows different rules and properties than regular multiplication.

3. What are some applications of multiplication modulo algebra?

Multiplication modulo algebra has various applications in fields such as cryptography, computer science, and abstract algebra. It is used in algorithms for generating secure encryption keys, error-correcting codes, and in solving problems related to number theory.

4. Can any number or polynomial be used as the modulus in multiplication modulo algebra?

Not all numbers or polynomials can be used as the modulus in multiplication modulo algebra. For a number to be suitable, it must be relatively prime to the other operand. Similarly, for a polynomial to be suitable, it must be irreducible (cannot be factored into smaller polynomials) and relatively prime to the other operand.

5. What are the main properties of multiplication modulo algebra?

The main properties of multiplication modulo algebra include commutativity, associativity, distributivity, and the existence of multiplicative inverses. It also follows the law of congruence, which states that if two numbers or polynomials have the same remainder when divided by a given modulus, then their product will also have the same remainder when divided by that modulus.

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